A203621 Highly anti-imperfect numbers: numbers k that sets a record for the value of |sigma*(k)-k|, where sigma*(k) is the sum of the anti-divisors of k.

1, 2, 7, 10, 13, 17, 22, 27, 28, 32, 38, 45, 52, 60, 63, 67, 77, 95, 105, 130, 137, 143, 157, 158, 175, 193, 203, 247, 297, 315, 357, 423, 462, 472, 473, 578, 675, 682, 742, 770, 787, 1012, 1138, 1215, 1417, 1463, 1732, 1957, 2047, 2048, 2327, 2363, 2632

Offset

1,2

Comments

Anti-imperfect numbers are anti-deficient numbers or anti-abundant numbers.

Links

Amiram Eldar, Table of n, a(n) for n = 1..600 (terms 1..100 from Paolo P. Lava)

Example

n=1. Anti-divisors: 0. |0-1|=1
n=2. Anti-divisors: 0. |0-2|=2
n=3. Anti-divisors: 2. |2-3|=1 less than 2: 3 is not in the sequence.
n=4. Anti-divisors: 3. |3-4|=1 less than 2: 4 is not in the sequence.
n=5. Anti-divisors: 2,3. |5-3|=2 equal to the maximum: 5 is not in the sequence.
n=6. Anti-divisors: 4. |4-6|=2 equal to the maximum: 6 is not in the sequence.
n=7. Anti-divisors: 2,3,5. |10-7|=3 new maximum: 7 is in the sequence.

Maple

P:=proc(i)
local a, k, n, s;
s:=0;
for n from 1 to i do
  a:=0;
  for k from 2 to n-1 do if abs((n mod k)- k/2)<1 then a:=a+k; fi; od;
  if abs(n-a)>s then s:=abs(n-a); print(n); fi;
od;
end:
P(3000);

Mathematica

sig[n_] := Total[Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]]; d[n_] := Abs[sig[n] - n]; s = {}; dm = -1; Do[If[(d1 = d[n]) > dm, dm = d1; AppendTo[s, n]], {n, 1, 2700}]; s (* Amiram Eldar, Jan 13 2022 after Michael De Vlieger at A066417 *)

Crossrefs

Cf. A066417, A074918, A192267, A192268.

Sequence in context: A277551 A029904 A026364 * A297832 A003158 A130336

Adjacent sequences: A203618 A203619 A203620 * A203622 A203623 A203624

Keyword

nonn

Author

Paolo P. Lava, Jan 04 2012

Status

approved